论文标题
关于基本代数超级几何的注释。希尔伯特和皮卡德超级化学
Notes on fundamental algebraic supergeometry. Hilbert and Picard superschemes
论文作者
论文摘要
这些注释旨在提供对代数超级几何的某些基本方面的完整，系统的说明，即，许多古典概念，技术和结果的Superschemes的几何形状构成了代数几何学的一般骨干，其中大多数来自Grethendieck的工作。特别是，我们扩展到代数超级几阶，例如投影和适当的形态，辅助学，矢量和投影捆绑包，辅助基础变化，半持续定理，相对偶尔，castelnuovo-mumford的规律性，扁平化，希尔伯特（Hilbert），希尔伯特（Hilbert）和坦率的态度（toical flate cooly of Serige cooly sergectient）（castelnuovo-mumford）的cositectient（Subly Secartiest），Subary Serigent contectient（等等。在其他地方可能会发现一些结果，尤其是Moosavian和Zhou最近的预印本有些重叠。 However, many techniques and constructions are presented here for the first time, notably, a first development of Grothendieck relative duality for proper morphisms of superschemes, the construction of the Hilbert superscheme in a more general situation than the one already known (which in particular allows one to treat the case of sub-superschemes of supergrassmannians), and a rigorous construction of the Picard superscheme for a locally superprojective Noetherian Superschemes具有几何积分纤维的态度。此外，即使仅限于普通计划，这里给出的一些证据也是新的。在最后一节中，我们构建了一个周期图，从适当和平滑的超级弯曲的模量的开放式替代到主要极化的Abelian Superekemes的模量堆栈。
These notes aim at providing a complete and systematic account of some foundational aspects of algebraic supergeometry, namely, the extension to the geometry of superschemes of many classical notions, techniques and results that make up the general backbone of algebraic geometry, most of them originating from Grothendieck's work. In particular, we extend to algebraic supergeometry such notions as projective and proper morphisms, finiteness of the cohomology, vector and projective bundles, cohomology base change, semicontinuity theorems, relative duality, Castelnuovo-Mumford regularity, flattening, Hilbert and Quot schemes, faithfully flat descent, quotient étale relations (notably, Picard schemes), among others. Some results may be found elsewhere, and, in particular, there is some overlap with a recent preprint by Moosavian and Zhou. However, many techniques and constructions are presented here for the first time, notably, a first development of Grothendieck relative duality for proper morphisms of superschemes, the construction of the Hilbert superscheme in a more general situation than the one already known (which in particular allows one to treat the case of sub-superschemes of supergrassmannians), and a rigorous construction of the Picard superscheme for a locally superprojective morphism of noetherian superschemes with geometrically integral fibres. Moreover, some of the proofs given here are new as well, even when restricted to ordinary schemes. In a final section we construct a period map from an open substack of the moduli of proper and smooth supercurves to the moduli stack of principally polarized abelian superchemes.
